The Traveling Salesman Problem with Few Inner Points

We study the traveling salesman problem (TSP) in the 2-dimensional Euclidean plane. The problem is NP-hard in general, but trivial if the points are in convex position. In this paper, we investigate the influence of the number of inner points (i.e., points in the interior of the convex hull) on the computational complexity of the problem. We give two simple algorithms for this problem. The first one runs in O(k!kn) time and O(k) space, and the second runs in O(2 k k 2 n) time and O(2 k kn) space, when n is the total number of input points and k is the number of inner points. Hence, if k is taken as a parameter, this problem is fixed-parameter tractable (FPT), and also can be solved in polynomial time if k=O(log n). We also consider variants of the TSP such as the prize-collecting TSP and the partial TSP in this setting, and show that they are FPT as well.

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